Julen Lopetegui replaced Moyes at the London Stadium this summer, but is under huge pressure.
Fail to get a result against Wolves on Monday night, in a game dubbed 'El Sackico', and he would be unlikely to last much longer.
Now it is understood that Moyes would be open to heading back.
West Ham have had discussions with former Juventus boss Max Allegri, but he would only want to take over next summer.
The Hammers could turn to a caretaker manager should they sack Lopetegui.
READ MORE IN FOOTBALL
Palmer embodies many things Tottenham desperately lack and Chelsea's title bid is real
Chelsea fans spot unseen moment where Nicolas Jackson stops teammate getting yellow card
A move to bring in a caretaker prior to Allegri's arrival would open the door to Moyes' return.
The 61-year-old has taken charge of 261 matches across his two previous tenures.
He won 112 of those matches and guided the Hammers to consecutive top seven finishes.
Those were followed by a Europa Conference League triumph in 2023, ending a 43-year wait for a major trophy.
Most read in Football
Michail Antonio faces agonising 12 month wait to see if he can return to football
Rooney hires former Man United coach after Plymouth assistant leaves for personal reasons
Palmer embodies many things Tottenham desperately lack and Chelsea's title bid is real
Postecoglou admits he can't ask for patience as unique stat highlights defensive woes
However, Moyes left the club when his contract expired at the end of last term following a ninth place finish.
He later revealed he had been offered a new deal earlier in the campaign before it was rescinded by owner David Sullivan.
Under Lopetegui, the Hammers have won four out of 14 Premier League matches.
Their defeat to Leicester last Tuesday leaves the Spaniard's side 14th in the table ahead of the visit of Wolves.
The weekend's results have seen their gap to the relegation zone remain at six points.
Wolves could half that deficit with a win at the London Stadium, though.
West Ham fan wants David Moyes BACK at the club as they call for Julen Lopetegui to be SACKED